Is 0! = 10! = 1 because there is only one way to do nothing?

The proof for 0!=1 was already asked at here. My question, yet, is a bit apart from the original question. I’m asking whether actually 0!=1 is true because there is only one way to do nothing or just because of the way it’s defined.

Answer

Yes, precisely there is a unique function \emptyset \to \emptyset (with empty graph), which happens to be a bijection (\operatorname{id}_\emptyset). Note, that n! is the number of bijections \{1,\dots, n\}\to \{1,\dots,n\}.

Attribution
Source : Link , Question Author : Ali Abbasinasab , Answer Author : Stefan Perko

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